Abstract
This paper is aimed at establishing the generalized forms of Riemann-Liouville fractional inequalities of the Hadamard type for a class of functions known as strongly exponentially --convex functions. These inequalities provide some general formulas from which one can get associated inequalities for various types of exponentially convex and strongly convex functions. Refinements of well-known inequalities are also deducible from the established theorems.
1. Introduction
The notion of convexity is utilized significantly for finding solutions of essential mathematical problems in subjects of science and engineering. Leading with major developments in several branches of mathematics, convexity made its way in statistics, geometric function theory, graph theory, and economics. In recent decades, classes of functions related to convex functions are frequently used in proving new fractional integral inequalities in the form of numerous refinements and generalizations of classical inequalities.
Let be an interval of real numbers. A function satisfying for all and , is called convex function.
A convex function satisfies the well-known Hadamard inequality:
If is concave function, then, (1) holds in a reverse order. The inequality (1) had/has been studied by many researchers and consequently obtained a lot of its variants by introducing new classes of functions. For example, in [1], it is studied for -convex functions; in [2], it is studied for -convex functions; in [3, 4], it is studied for harmonically convex functions; in [5], it is studied for strongly harmonically convex and strongly -convex functions. Our goal in this paper is to study the inequality (1) for strongly exponentially --convex functions.
Definition 1 (see [6]). A function is called strongly exponentially (, -)--convex with modulus , if is nonnegative and holds, while is an interval containing and is a nonnegative function along with , , , and .
By using (2), one can find various classes of functions closely related with the convex function and strongly convex functions already defined by different authors. Strongly convex functions provide the refinements of convex functions.
In [2], Theorem 5, if we take , , and , then, we have the following theorem.
Theorem 2. Let be a positive function such that . If is a -convex function on , . Then, the following integral inequality holds:
Our aim in this paper is the derivation of compact forms of Hadamard-type inequalities for strongly exponentially --convex functions via Riemann-Liouville fractional integrals involving monotone functions. The established formulas will generate Hadamard-type inequalities for fractional Riemann-Liouville integrals which have been published by various authors in the recent past (see Remarks 11 & 23). Also, Hadamard-type inequalities are deducible for some new classes of functions (see Corollaries 12–32). In the following, we give the definition of Riemann-Liouville fractional integrals:
Definition 3. Let . Then, Riemann-Liouville fractional integral operators of order for a function , where , are given by Next, we give Hadamard-type inequalities via Riemann-Liouville fractional integrals of convex functions as follows:
Theorem 4 (see [7]). Let be a positive function with and . If is a convex function on , then, the following fractional integral inequality holds: with .
Theorem 5 (see [8]). With the assumptions given in Theorem 4, one can have the fractional integral inequality as follows: with .
Theorem 6 (see [7]). Let be a differentiable mapping on with . If is convex on , then, the following fractional integral inequality holds:
The definition of -fractional Riemann-Liouville integral operators is given as follows:
Definition 7 (see [9]). Let , . Then, -fractional Riemann-Liouville integrals for a function of order where are given by where is defined as follows:
The generalized form of Riemann-Liouville fractional integrals is given in the following definition:
Definition 8 (see [10]). Let . Also, let be an increasing and positive monotone function on , having a continuous derivative on . The left-sided and right-sided fractional integrals of a function with respect to another function on of order where are given by
The definition of the -analogue of the abovementioned definition is given as follows:
Definition 9 (see [11]). Let be the same as in the abovementioned definition. Then, for , the -analogue of (10) and is given by Using the fact that in (10) after replacing by , we get For further detailed study on fractional integrals, we refer the readers to [12, 13]. In the next section, we formulate the Hadamard-type inequalities for strongly exponentially --convex function via integrals (10) which are compact forms of a plenty of well-known Hadamard-type inequalities holding for classes of convex, strongly convex, and exponentially convex functions. Specifically, one can have refinements of the inequalities proved in recent decades. Several special case inequalities in the form of corollaries are also given.
2. Main Results
We will use the following notations for terms which will appear frequently in the results of this section
Theorem 10. Let , , range be the positive functions such that , and be a differentiable and strictly increasing. If is strongly exponentially --convex function on such that and , then, for , the following fractional integral inequalities hold: (i)If ,with , , for all and (ii)For , one can havewith , , for all and
Proof. The following inequality holds for a strongly exponentially --convex function
By setting , in (18) and then integrating on after multiplying with , one can get
Setting and in (19) and multiplying by , after applying Definition 3, the following inequality can be obtained:
Now, by using definition of strongly exponentially --convex function for and then integrating the resulting inequality on after multiplying with , one can get
Again, using substitution as considered in (20) leads to the second inequality of (14)
(ii) The proof is followed on same lines as the proof of (i)
Remark 11. The aforementioned version of the Hadamard inequalities gives (i) [4], Theorem 4 for , , , and ; (ii) [3], Theorem 2.4 for , , , and ; (iii) [14], Theorem 3.10 for and ; (iv) [15], Corollary 2.2 for , , and ; (v) Theorem 2 for , , and ; (vi) [16], Theorem 2.1 for , , and ; (vii) [14], Theorem 2.2 for and ; (viii) Theorem 1 for , , and ; (ix) [17], Theorem 2.1 for , , , , and ; (x) [1], Theorem 2.1 for , , , and ; and (xi) [6], Theorem 3 for . Moreover, the refinements of all the deduced results will occur for .
Corollary 12. For , one can have for the strongly --convex function the following fractional integral inequality:
Proof. The abovementioned inequality can be deduced by setting in (14).
(ii) For , one can have for the strongly --convex function the following fractional integral inequality:
Proof. The abovementioned inequality can be deduced by setting in (16).
Corollary 13. For , one can have for the strongly exponentially --convex function the following fractional integral inequality:
Proof. The abovementioned inequality can be deduced by setting in (14).
(ii) For , one can have for the strongly exponentially --convex function the following fractional integral inequality:
Proof. The abovementioned inequality can bed deduced by setting in (16).
Corollary 14. For , one can have for the strongly exponentially --Godunova-Levin function the following fractional integral inequality:
Proof. The abovementioned inequality can be deduced by setting and in (2.1).
(ii) For , one can have for the strongly exponentially --Godunova-Levin function the following fractional integral inequality:
Proof. The abovementioned inequality can be deduced by setting and in (16).
Corollary 15. For , one can have for the strongly exponentially --convex function in the third sense the following fractional integral inequality:
Proof. The abovementioned inequality can be deduced by setting and in (14).
(ii) For , one can have for the strongly exponentially --convex function in third sense the following fractional integral inequality:
Proof. The abovementioned inequality can be deduced by setting and in (16).
Corollary 16. For , one can have for the strongly exponentially --convex function the following fractional integral inequality:
Proof. The abovementioned inequality can be deduced by setting in (14).
(ii) For , one can have for the strongly exponentially --convex function the following fractional integral inequality:
Proof. The abovementioned inequality can be deduced by setting in (16).
Corollary 17. For , one can have for the strongly exponentially -convex function the following fractional integral inequality:
Proof. The abovementioned inequality can be deduced by setting and in (14).
(ii) For , one can have for strongly exponentially -convex function the following fractional integral inequality
Proof. The abovementioned inequality can be deduced by setting and in (16).
Corollary 18. For the strongly exponentially -HA-convex function, the following inequality holds:
Proof. The abovementioned inequality can be deduced by setting in (16).
Corollary 19. For the strongly exponentially -HA-convex function, the following inequality holds: